ALBANY — Gov. Cuomo routed a recalcitrant Legislature to secure a decisive first victory yesterday by delivering an early budget deal that would erase a $10 billion gap and shake up state government without borrowing or tax hikes.

It calls for a year-to-year spending slash of roughly $3.4 billion, or about 2 percent, the first such cut since the mid-1990s. More significantly, it would cap growth in education and Medicaid spending at 4 percent in 2012, helping to trim an expected 2012 deficit from $15 billion to $2 billion.

“It’s a new day in New York,” said Cuomo, who was flanked by legislative leaders. “We set out to build a new New York, and this is the first step down that road.”

The plan would empower the governor to close as many as six prisons and cut the number of state agencies by 20 percent. It would merge the Banking and Insurance departments and cut projected Medicaid growth by $2.8 billion.

It would not renew an income-tax surcharge on the rich, which Silver had championed as a way to offset massive cuts.

Silver said, “This is a sobering budget unquestionably; government had to tighten its belt. The sole consolation is that working with the governor, we were able to achieve critical restorations.”

The Legislature reduced Cuomo’s historic $1.5 billion year-to-year cut to education aid by about $271 million and added another $210 million to various college, social-services and agriculture programs.

Mayor Bloomberg, however, decried state leaders for their failure to restore $302 million in local aid suspended last year by then-Gov. David Paterson and eliminated by Cuomo. The mayor said not enough education aid was restored to avoid teacher layoffs.

“Make no mistake: the final budget still cuts New York City more than ever before,” Bloomberg said.

Though numbers were still being sorted out, budget officials said the final version would be only $250 million larger than Cuomo’s original proposal. Some new spending would be offset by new cuts, including $170 million to the state’s judiciary, which had initially refused to reduce its spending.

One Democratic lawmaker described Cuomo’s victory as “unprecedented” and a blow to Silver.

“He’s going to get largely his budget, largely what he wanted, and Shelly’s going to be the loser,” the lawmaker said.